"Does not trouble the reason"
I would say it means "does not pose a problem with someone's faculty of reason", "reason" being the desire and need to rationally explain what one is faced with.
You can understand "trouble" as in "troubled his conscience" or "troubled his mind", "trouble in mind", etc.
"trouble the reason" doesn't seem to turn up anything if you search for it.
The reason may be because we seldom say it like that but rather "trouble my reason", "trouble his reason", and so son.
Here the author is talking about reason as something independent in a medical context.
If you google "troubled my reason" or "troubled his reason", you do get some usage hits, though still relatively few.